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8/9/2019 Receuil Problems in Mathematical Analysis D
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G. Baranenkov*
B. Drmidovich
V.
Efimenko,
S.
Kogan,
G.
Lunts>>
E.
Porshncva,
E.
bychfia,
S.
frolov,
/?.
bhostak,
A.
Yanpolsky
PROBLEMS
IN
MATHEMATICAL
ANALYSIS
Under the
editorship
of
B.
DEMIDOVICH
Translated
from
the
Russian
by
G.
YANKOVSKV
MIR PUBLISHERS
Moscow
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T.
C.
BapaneHKoe.
B.
77
AeMudoeuH,
B.
A
C.
M.
Koean,
r
Jl
JJyHit,
E
noptuneea,
E H.
Ctweea,
C. B.
P.
fl.
UlocmaK,
A.
P.
SAflAMM
H
VnPA)KHEHHfl
no
MATEMATM
H
ECKOMV
AHAJ1H3V
I7od
B.
H.
AE
rocydapcmeeHHoe
a
M
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OUP-43
30-1.71
5,000
OSMANIA
UNIVERSITY
LIBRARY
Call
No.
g>^^
Accession
No.
Author
This
book
should
be
returned
on
or
before
the
date
last
marked
below
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M
I
R
PUBLISH
P.
KS
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Contents
Sec.
6.
Integrating
Certain Irrational
Functions 125
Sec
7.
Integrating
Trigoncrretric
Functions
128
Sec. 8
Integration
of
Hyperbolic
Functions
133
Sec
9.
Using
Ingonometric
and
Hyperbolic
Substitutions
for
Finding
integrals
of
the
Form
f
R
(x, ^a^+
bx
+
c)
dx, Where
R
is
a
Ra-
tional
Function
133
Sec
10
Integration
of
Vanou*
Transcendental
Functions
135
Sec
11
Using
Reduction
Formulas
135
Sec.
12.
Miscellaneous
Examples
on
Integration
136
Chapter
V DEFINITE
INTEGRALS
Sec. 1.
The
Definite
Integral
as
the
Limit
of
a
Sum
138
Sec
2
Evaluating
Ccfirite
Integrals
by
Means of
Indefinite
Integrals
140
Sec.
3
Improper
Integrals
143
Sec
4
Charge
of
Variable
in
a
Definite
Integral
146
Sec.
5.
Integration
by
Parts
149
Sec
6
Mean-Value
Theorem
150
Sec.
7.
The
Areas
of Plane
Figures
153
Sec
8.
The Arc
Length
of
a
Curve
158
Sec
9
Volumes
of
Solids
161
Sec
10
The
Area
of
a
Surface
of
Revolution
166
Sec
11 torrents
Centres
of
Gravity
Guldin's
Theorems
168
Sec
12.
Applying
Definite
Integrals
to
the
Solution of
Physical
Prob-
lems
173
Chapter
VI.
FUNCTIONS
OF
SEVERAL
VARIABLES
Sec.
1.
Basic
Notions
180
Sec. 2.
Continuity
184
Sec
3
Partial
Derivatives
185
Sec
4 Total
Differential of
a
Function
187
Sec
5
Differentiation of
Composite
Functions
190
Sec.
6.
Derivative in
a
Given Direction
and the
Gradient
of a Function
193
Sec.
7
HigKei
-Order
Derivatives
and
Differentials
197
Sec 8
Integration
of
Total
Differentials
202
Sec
9
Differentiation of
Implicit
Functions
205
Sec
10
Change
of
Variables
.211
Sec. 11.
The
Tangent
Plane
and
the
Normal to
a
Surface
217
Sec
12
Taylor's
Formula for a
Function of Several
Variables
.
.
.
220
Sec.
13 The
Extremum
of a
Function
of
Several Variables
....
222
Sec
14
Firdirg
the
Greatest
and
*
tallest Values
of
Functions
.
. 227
Sec
15
Smcular
Points
of
Plane
Curves
230
Sec 16
Envelope
.
.
232
Sec.
17.
Arc
Length
o
a
Space
Curve
234
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Contents
Sec.
18.
The
Vector
Function
of
a Scalar
Argument
235
Sec.
19
The
Natural
Trihedron
of
a
Space
Curve
238
Sec.
20.
Curvature
and
Torsion
of
a
Space
Curve
242
Chapter
VII. MULTIPLE
AND
LINE
INTEGRALS
Sec. 1
The
Double
Integral
in
Rectangular
Coordinates
246
Sec.
2
Change
of Variables
in
a
Double
Integral
252
Sec.
3.
Computing
Areas
256
Sec.
4.
Computing
Volumes
258
Sec.
5.
Computing
the
Areas
of Surfaces 259
Sec.
6
Applications
of
the
Double
Integral
in
Mechanics
230
Sec. 7.
Triple
Integrals
262
Sec.
8.
Improper Integrals
Dependent
on a
Parameter.
Improper
Multifle
Integrals
269
Sec.
9
Line
Integrals
273
Sec.
10.
Surface
Integrals
284
Sec.
11.
The
Ostrogradsky-Gauss
Formula
286
Sec.
12.
Fundamentals
of
Field
Theory
288
Chapter
VIII.
SERIES
Sec.
1.
Number
Series
293
Sec.
2.
Functional Series
304
Sec.
3.
Taylor's
Series
311
Sec.
4.
Fourier's
Series
318
Chapter
IX
DIFFERENTIAL
EQUATIONS
Sec.
1.
Verifying
Solutions.
Forming
Differential
Equations
of Fami-
lies
of
Curves.
Initial
Conditions
322
Sec.
2
First-Order Differential
Equations
324
Sec.
3.
First-Order
Diflerential
Equations
with
Variables
Separable.
Orthogonal
Trajectories
327
Sec.
4
First-Order
Homogeneous
Differential
Equations
330
Sec.
5.
First-Order
Linear Differential
Equations.
Bernoulli's
Equation
332
Sec.
6
Exact
Differential
Equations.
Integrating
Factor
335
Sec
7 First-Order
Differential
Equations
not Solved
for
the
Derivative
337
Sec.
8.
The
Lagrange
and
Clairaut
Equations
339
Sec.
9. Miscellaneous
Exercises
on
First-Order
Differential
Equations
340
Sec.
10.
Higher-Order
Differential
Equations
345
Sec.
11.
Linear
Differential
Equations
349
Sec.
12.
Linear
Differential
Equations
of Second Order
with
Constant
Coefficients
351
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8
Contents
Sec.
13.
Linear
Differential
Equations
of Order
Higher
than
Two
with
Constant
Coefficients
356
Sec 14.
Euler's
Equations
357
Sec
15.
Systems
of
Differential
Equations
359
Sec.
16.
Integration
of Differential
Equations
by
Means
of Power
Se-
ries
361
Sec
17. Problems
on
Fourier's
Method
363
Chapter
X.
APPROXIMATE
CALCULATIONS
Sec.
1
Operations
on
Approximate
Numbers
367
Sec.
2.
Interpolation
of
Functions 372
Sec.
3.
Computing
the^Rcal
Roots
of
Equations
376
Sec.
4
Numerical,
Integration
of
Functions 382
Sec. 5.
Nun
er:ca1
Integration
of
Ordinary
DilUrtntial
Equations
.
.
384
Sec.
6.
Approximating
Ftuncr's
Coefficients
3>3
ANSWERS
396
APPENDIX
475
I.
Greek
Alphabet
475
II.
Some Constants
475
III.
Inverse
Quantities,
Powers, Roots,
Logarithms
476
IV
Trigonometric
Functions
478
V.
Exponential,
Hyperbolic
and
Trigonometric
Functions
479
VI.
Some Curves
480
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PREFACE
This collection of
problems
and
exercises
in
mathematical
anal-
ysis
covers
the
maximum
requirements
of
general
courses
in
higher
mathematics
for
higher
technical
schools.
It
contains
over
3,000
problems sequentially
arranged
in
Chapters
I
to
X
covering
all
branches
of
higher
mathematics
(with
the
exception
of
ana-
lytical
geometry)
given
in
college
courses.
Particular
attention
is
given
to
the
most
important
sections
of
the
course
that
require
established
skills
(the
finding
of
limits,
differentiation
techniques,
the
graphing
of
functions,
integration
techniques,
the
applications
of definite
integrals,
series,
the solution
of
differential
equations).
Since some
institutes
have extended
courses
of
mathematics,
the
authors
have
included
problems
on
field
theory,
the
Fourier
method,
and
approximate
calculaiions.
Experience
shows that
the number
of
problems
given
in
this book
not
only
fully
satisfies
the
requireiren
s of
the
student,
as
far
as
practical mas ering
of
the
various
sections
of
the
course
goes,
but
also
enables
the
in-
structor
to
supply
a
varied choice of
problems
in
each
section
and
to
select
problems
for
tests
and
examinations.
Each
chap.er
begins
with
a brief
theoretical
introduction
that
covers
the
basic
definitions
and formulas
of that section
of
the
course.
Here the
most
important
typical problems
are
worked
out
in full.
We
believe that this will
greatly
simplify
the
work
of
the
student.
Answers
are
given
to
all
computational
problems;
one
asterisk
indicates
that
hints
to the
solution
are
given
in
the
answers,
two
asterisks,
that
the
solution
is
given.
The
problems
are
frequently
illustrated
by
drawings.
This
collection
of
problems
is the
result
of
many years
of
teaching
higher
mathematics in the technical
schools of
the
Soviet
Union.
It
includes,
in
addition
to
original
problems
and
exam-
ples,
a
large
number
of
commonly
used
problems.
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Chapter
I
INTRODUCTION
TO
ANALYSIS
Sec.
1. Functions
1.
Real
nurrbers.
Rational
and
irrational
numbers
are
collectively
known
as
real
numbers
The
absolute
value
of
a real
number
a
is
understood
to
be
the
nonnegative
number
\a\
defined
by
the
conditions'
\a\=a
if
a^O,
and
|aj
=
a if a
< 0.
The
following
inequality
holds for
all
real
numbers
a
ana
b:
2.
Definition
of
a
function.
If
to
every
value*)
of
a
variable
x,
which
belongs
to
son.e collection
(set)
E,
there
corresponds
one and
only
one
finite
value
of
the
quantity
/,
then
y
is
said
to
be
a
function
(single-valued)
of
x
or
a
dependent
tariable defined on
the
set
E.
x is
the a
r
gument
or
indepen-
dent
variable
The
fact
that
y
is
a Junction of
x
is
expressed
in brief form
by
the notation
y~l(x)
or
y
=
F
(A),
and
the
1'ke
If to
every
value of x
belonging
to
some
set
E
there
corresponds
one
or
several
values
of
the
variable
/y,
then
y
is
called
a
multiple-
valued
function
of
x
defined
on
E.
From
now
on
we
shall use
the
word
function
only
in
the
meaning
of
a
single-valued
function,
if not otherwise
stated
3
The
domain
of
definition
of
a
function. The
collection
of
values
of
x
for
which the
given
function
is defined
is
called
the
domain
of
definition
(or
the
domain)
of
this
function.
In the
simplest
cases,
the domain
of
a
function is
either
a
closed interval
[a.b\,
which
is
the
set
of
real
numbers
x
that
satisfy
the
inequalities
a^^^b,
or
an
open
intenal
(a.b),
which
:s the
set
of
real
numbers
that
satisfy
the
inequalities
a
<
x
<
b.
Also
possible
is a
more
com-
plex
structure of
the
domain of definition
of
a
function
(see,
for
instance,
Prob-
lem
21)
Example
1.
Determine
the
domain
of definition of the
function
1
Solution.
The
function
is
defined
if
x
2
-l>0,
that
is,
if
|x|>
1.
Thus,
the
domain
of
the
function is
a
set
of
two inter-
vals:
oo
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12
Introduction
to
Analysis \Ch.
I
then
the function
x
=
g(y),
or,
in
standard
notation,
y=g(x),
is
the
inverse
of
y
=
f(x).
Obviously,
g[f(x)]s&x,
that
is,
the
function
f
(x)
is
the
inverse
of
g(x)
(and
vice
ve^sa).
In
He
fereia
case,
the
equation
y
f(x)
defines
a
multiple-valued
in-
verse function x
=
f~
}
(y)
such
that
y
==[[(-*
(y)\
for
all
y
that are
values
of
the
function
f
(x)
Lxanple
2.
Determine
the inverse
of the
function
y=l-2-*.
(1)
Solution.
Solving
equation
(1)
for
x,
we
have
2-*=l
y
and
log(l-y)
*
log
2
j
'
w
Obviously,
the
domain
of
Definition
of
the function
(2)
is
oo
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Sec
1]
Functions
13
8. Find
the
rational
integral
function
f(x)
of
degree
two,
if
=
and
/(3)
=
5.
9.
Given
that
f(4)
=
2,
/(5)
=
6.
Approximate
the value
of
/(4,
3)
if
we
consider
the
function
/
(x)
on
the
interval
linear
(linear interpolation
of
a
function).
10.
Write the
function
0,
if
as
a
single
formula
using
the
absolute-value
sign.
Determine
the domains
oi definition
of
the
following
functions:
11.
a)
y
=
x+\;
16.
y
=
x
-
17.
/
=
lo
13.
a),=
?E2L
b)
(/
=
*VV-2.
19.
t/=
14**.
=1/2 +
*
**.
-
21. Determine
the
domain
of
definition
of
the
function
y
=
|/sin
2x.
22.
f(jc)
=
2A:
4
SA;'
5x
8
+
6A:
10.
Find
(-*)l
and
^(^)
=
23.
A
function
f
(x)
defined in
a
symmetric
region
/
is
called
euen
if
f(
x)
=
f(x)
and
orfd
if
/( x)
=
f(x).
Determine
which
of
the
following
functions
are
even
and
which
are
odd:
e)
24.
Prove
that
any
function
f(x)
defined
in the
interval
/
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14
In
t
roduction
to
Analysis
[Ch.
1
25.
Prove
that
the
product
of two
even functions or
of
two
odd
functions
is
an
even
function,
and
that
the
product
of
an even
function
by
an odd
function
is
an
odd function.
26.
A
function
f
(x)
is
called
periodic
if
there
exists a
positive
numter
T
(the
period
of
the
function)
such
that
f(x+
T)^f(x)
for
all valves of
x
within
the
dcmain
of
definition of
f(x).
Determine uhich
of
the
following
functions
are
periodic,
and
for
the
periodic
functions find
their
least
period
T:
a)
/
(x)
=
10
sin
3
*,
d)
/
(x)
=
sin
1
*;
b)
/
(*)
=
a
sin
\K
+
b
cos
tar;
e)
/
(x)
=
sin
(J/*).
c)
27.
Express
the
length
of
the
segment
y
=
MN
and
the
area
S
of
the
figure
AMN
as
a function
of
x=AM
(Fig
1).
Construct
the
graphs
of
these
functions.
28.
The
linear
density
(that
is,
mass
per
unit
length)
of
a
rod
AB
=
l
(Fig.
2)
on
the
segments
AC
l^
CD
=
1
2
and
DB
=
l\
(/
t
+
l
t
+
/
3
=-
1)
AfsM
\
I f
is
equal
to
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Sec.
1]
_
Functions
_
15
34.
Prove that
if
f(x)
is an
exponential
function,
that
is,
/
(x)
=
a
x
(a
>0),
and the
numbers
x
v
*,,
x
t
form
an
arithmetic
progression,
then the
numbers
/(*,),
f (*
2
)
and
/(jcj
form
a
geo-
metric
progression.
35.
Let
Show that
36.
Let
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16
Introduction
to
Analysts
[Ch.
1
42.
Write as
a
single
equation
the
composite
functions
repre-
sented
as
a
series of
equalities:
a)
y
=
u*>
w
=
sin#;
b)
#
=
arctan,
u =
Yv,
y
=
log#;
w,
if
t/0;
*_ .
43.
Write,
explicitly,
functions
of
y
defined
by
the
equations:
a)
x
2
arc
cos
y
=
n;
b)
10*
+10'
=10;
c)
*
+
\y\
=
2y.
Find
the
domains
of
definition
of
the
given
implicit
functions.
Sec,
2.
Graphs
of
Elementary
Functions
Graphs
of
functions
#
=
/(*)
are
mainly
constructed
by
marking
a
suffi-
ciently
dense
net
of
points
Ai
/(*,-,
//),
where
*/,
=
/
(*,-)
(/
=
0,
1,
2,
...)
and
by
connecting
the
points
with
a line that
takes
account
of
intermediate
points.
Calculations
are
best done
by
a slide
rule.
Fig.
3
Graphs
of
the basic
elementary
functions
(see
Ap pendix VI)
are
readily
learned
through
their
construction.
Proceeding
from
the
graph
of
y
=
f(x),
(T)
we
get
the
graphs
of
the
following
functions
by
means of
simple
geometric
constructions:
1)
0i
=
M*)
js
th
*
mirror
image
of
the
graph
T
about
the
*-axis;
2)
0i=/(
*)
is
the
mirror
image
of
the
graph
F
about
the
#-axis;
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Sec.
2]
Graphs
of
Elementary
functions
17
3)
#i
=
/(*)
is
the
F
graph
displaced
along
th?
x-axis
by
an
amount
a;
4)
1/4
=
&
+
/(*)
is
the
F
graph
displaced
along
Uve
(/-axis
by
an
amount
(Fg. 3).
Example.
Construct the
graph
of the
function
Solution. The
desired
line
is
a
sine curve
y
=
sinx
displaced along
the
*-axis
to
the
right
by
an
amount
-j
(Fig.
4)
Y
Fig.
4
Construct the
graphs
of
the
following
linear
functions
(straight
lines):
44.
y
=
kx,
if fc
=
0,
1,
2,
1/2,
-1,
-2.
45.
i/
=
x+
6,
if 6
=
0,
1,
2,
1,
2.
46.
0=1.
5*
+2.
Construct
the
graphs
of
rational
integral
functions
of
degree
two
(parabolas).
47.
y=--ax
3
,
if
a=l, 2,
1/2,
1, 2,
0.
48.
//
=
*'-{-
c,
if
c=0,
1,
2,
1.
49.
,/=(*-*)',
if
*.
=
f 1,
2,
-1.
50.
y
=
y,
4
(x-l)\
if
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18
_
Introduction
to
Analysis
_
[Ch.
1
59.
-.
61*.
y=
62
*-
*-
Construct
the
graphs
of
the
fractional rational functions:
63.
=,.
J^l-A
2
__
78*.
y
=
+
x
y
^^
(cissoid
of Diodes).
79.
r/
==
x
1/25
x
2
.
Construct
the
graphs
of the
trigonometric
functions:
80*.
y
=
sinx.
83*.
*/=-cotjc.
81*.
y
=
cosx.
84*.
y
=
sec
x.
82*.
/=-tanx.
85*.
y
=
cosec
x.
86.
{/-/4sinx
f
if
/4
=
1,
10,
1/2,
2.
87*.
y^smnx,
if
n=l,
2, 3,
1/2.
/ \ * rv
Jl
3ll
ft
88.
y=
sin(x
cp),
if
9
=
0,
-J-,
-j-
n
~T*
89*.
y
=
5sin(2x
3).
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Sec.
2] Graphs
of
Elementary
functions
19
Construct
the
graphs
of the
exponential
and
logarithmic
func-
tions:
101.
/
=
a
x
,
if
a
=
2,
l
f
(?(e
=
2,
718
...)*).
102*.
y
=
\og
a
x,
if
a
=10,
2,
1,
*.
103*.
y
=
sinhx,
where
sinhx=l/2(e
x
e
-x
).
104*.
f/
=
coshx,
where
coshx
=
105*.
f/
=
tanhx,
where
106.
0=10*.
107*.
y=-e~*
2
(probability
curve).
108.
^
=
2
*
3
.
113.
y
=
109.
//-logx
2
.
114.
(/=--
110.
y-=log
2
A:.
115.
{/
=
111.
//=-log(logx).
116.
t/
=
log (cosx).
112.
/y==rV--
117
-
^
=
2-^
sin*.
log
X
Construct the
graphs
of
the
inverse
trigonometric
functions?
118*.
y--=arc
sin*. 122.
#
=
arcsin~.
x
119*.
j/
=
arccosx.
123.
#
=
arc
cos--.
120*.
#
=
arc
tan*.
124.
^
=
A:
+
arc
cot
x.
121*,
(/=
arc
cot
x.
Construct
the
graphs
of
the
functions:
125.
y=\x\.
126.
y
=
^(x
+
\x\).
127.
a)
y
=
x\x\\
b)
y
=
log^^l
x
\-
128.
a)
t/= sinA:+
|
sin
jt|;
b)
f/=
sin
x
|
sinx|.
3
x
2
when
|jc|<
1.
129.
^-<
_l.
whcn
*)
About
the
number
*
see
p.
22
for
more
details.
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20
Introduction
to
Analysis
[C/i.
/
130.
a)
#=[*],
b) y
=
x[x],
where
[x]
is
the
integral
part
of
the number
x,
that
is,
the
greatest
in.eger
less
than
or
equal
to x.
Construct
the
graphs
of
the
following
functions
in
the
polar
coordinate
system
(r,
cp)
(r^O):
131.
r
=
l.
132*.
f
=
7r
(spiral
of
Archimedes).
133*.
/-
=
(logarithmic
spiral).
134*. r
=
(hyperbolic
spiral).
135.
r
=
2cosip
(circle).
136.
'
=
-^-
(straight
line).
137.
/-
=
sec*y
(parabola).
138*.
r=10sin3(p
(three-leafed rose)
139*.
r
=
a(l fcoscp)
(a>0)
(cardioid).
143*.
r
I
=
a
2
cos2(p
(a>0)
(lemniscate).
Cjnstruct
the
graphs
of
the functions
represented
parametri-
cally:
141*.
x
=
t\
y
=
t*
(semicubical parabola).
142*.
*=10
cos/,
y=sin/
(ellipse).
143*. *=10cos
3
/,
y=
10
sin
1
/
(astroid).
144*.
jc
=
a(cos/-f
/
sin/),
t/
=
a(sm
/
/cos/)
(involute
of
a
circle).
145*.
^
=
^3,
J/
=
rTT'
^
0//wm
^
Descartes).
146
'
^'
/==
147.
xasfc'-t^-
1
,
y
=
2
t
2-
t
(branch
of
a
hyperbola).
143.
jc
=
2cos
f
f f
#
=
2
sin
2
/
(segment
of
a
straight
line).
149.
*-/-
/
2
,
y=t
2
t\
150.
x^a
t
(2
cos/
cos2/), */
=
a(2sin/
sin
2/)
(cardioid).
Cjnstruct 'the
graphs
of
the
following
functions
defined
implic-
itly:
151*.x
2
+
*/
2
=
25
(circle).
152.
xy--=
12
(hyperbola).
153*.
i/
2
=
2jc
(parabola).
154.
^1
+
^
=
155.
j/*
=
jc'(10
t 2
156*.
x
T
+
y
T
=;a
T
(astroid).
157*.
x
158.
*'
=
-
8/9/2019 Receuil Problems in Mathematical Analysis D
21/504
Sec.
2]
Graphs of
Elementary
Functions
21
159*.
|/V
+
y
2
=e
a
*
(logarithmic
spiral).
160*.
x*
+
y
8
3x//
=
(folium
of
Descartes).
161.
Derive
the
conversion
formula
Irom
the
Celsius
scale
(Q
to
the
Fahrenheit
scale
(F)
if it
is
known
that
0C
corresponds
to
32F
and 100C
corresponds
to
212F.
Construct
the
graph
of
the function
obtained.
162.
Inscribed
in
a
triangle
(base
6^=10,
altitude
h
=
6)
is a
rectangle
(Fig.
5).
Express
the
area
of the
rectangle
y
as a
func-
tion
of the
base
x.
Fig.
5
Fig
6
Construct
the
graph
of
this
function
and
find
its
greatest
value.
163. Given
a
triangle
ACB
with
BC
=
a,
AC
=
b
and
a
variable
angle
$
ACB
=
x
(Fig.
6).
Express
#
=
area
A
ABC
as
a
function
of
x.
Plot the
graph
of this
function
and
find
its
greatest
value.
164.
Give
a
graphic
solution
of
the equations:
a)
2x'
5x
+
2
=
0;
d)
I0'
x
=
x\
b)
x*
+
x
1=0;
e)
x=l
4
5sin;c;
c)
logJt
=
0.1jc;
f)
cot
x^x
(0
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22
Introdnction
to
Analysis
(C/t.
/
Sec.
3.
Lfmits
1.
The
limit
of
a
sequence.
The
number
a is
the
limit
of
a
sequence
*
x
lt
....
X0,
....
or
lim
x
n
a,
n
> oo
if
for
any
e>0
there
is
a
number N
=
N
(e)
such that
\x
n
a
|
<
e when
n>
N.
Example
1.
Show
that
Urn
5L
+
1.2.
(1)
n
-*
rt-r
1
Solution.
Form
the
difference
2*
+1
1
Evaluating
the absolute
value
of
this
difference,
we have:
1
-2
<
e,
(2)
if
n>-\
=
N
(e).
Thus,
for
every
positive
number
there
will
be
a
number
Af=
1
such
that
for
n
>
N we will
have
inequality
(2)
Consequently,
the
number
2
is
the limit
of
the
sequence
x
n
(2n-\-
l)/(n-fl),
hence,
formula
(1)
is
true.
2.
The
limit
of
a
function. We
say
that a function
/
(x)
-*-
A
as
x
-+
a
(A
and
a
are
numbers),
or
lim
f(x)
=
A,
x
-a
if
for
every
8
>
we
have
6
=
6
()
>
such
that
\f(x)A
|
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8/9/2019 Receuil Problems in Mathematical Analysis D
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Sec.
3]
Limits
23
For the
existence
of the limit
of
a function
/
(x)
as
jc-^o,
it
is
necessary
and
sufficient
to
have
the
following
equality:
/(a
O)-/
If
the
limits
lim
/,
(x)
and
lim
f
2
(x)
exist,
then
the
following theorems.
x ->a x
->
a
I
old:
1)
lim
[/,
(*)
+
/, (*)]
=
lim
/,
(x)
+
lim
f
,
(x);
x
-+
a
x
-+
a x
-*
a
2)
lim
[f,
(x)
f
2
(jc)J
=
lim
f,
(x).
lim
f
t
(x);
x-*a x
-*
a
x
-*
a
3)
lim
[f
, W/^ (JK)J
=
lim
/,
(x)l
lim
^
(x)
(lim
f
, (x)
^ 0).
x
-
o
*
-#
a
jc
-*
a
Jt
-^
a
The
following
two
limits are
frequently
used:
lim
ILi=i
AP
->-0
^
and
x
lim
[
1-J--L
)
=
lim
(l
+
a)
a
=*
=
2
71828
.
.
.
Example
2.
Find
the limits on
the
right
and
left of
the
function
/
(x)
=
arc
tan-
as
x
->-0.
Solution.
We have
arc tan
)=
x
J
2
and
lim
fa
x
.+
+o
\
f(-0)=
lim
faictanlW-4-
x->.
-
o
\
A:
/
2
Obviously,
the
function
/
(x)
in this
case
has
no limit as
x--0.
166.
Prove
that
as
n
*oo
the
limit
of
the
sequence
is
equal
to zero.
For
which
values
of
n
will
we
have
the
inequal-
ity
(e
is
an
arbitrary
positive
number)?
Calcula
e
numerically
for
a)
e
=
0.1;
b)
e
=
0.01;
c)
8
=
0.001
167.
Prove
that
the limit
of the
sequence
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24
_
Introduction
to
Analysis
_
[CH.
1
as rt
>oo
is
unity.
For
which
values
of
n>N
will we
have
the
inequality
K-l|
2
How
should
one
choose,
for
a
given
positive
number
e,
some
positive
number
6
so
that
the
inequality
|*
2
-4|
+0 X
+
+
00
X
->
00
170.
Find the limits
of
the
sequences:
a)
I
_
1 _
*
(-
1
)
1
*}
i,
2*3'
4
.
...
b)
1
1
1
_2n_
V)
1
3
'
5
'
'
'
'
'
2/i~l
'
'
'
'
c)
1/2;
1/2
1/2 ,
1/21/21/2 ,
.
.
.
;
d)
0.2,
0.23,
0.233,
0.2333,
.
.
.
Find
the
limits:
171.
Hm
a+4
+
l,+
...
n
*\*
*
*
172.
lirn
C
+ D (
+
)(>.
+
3)
fl -oo
n
173 Hm
l)
2n+11
2
J'
178*.
lim
n
-*
CD
-
8/9/2019 Receuil Problems in Mathematical Analysis D
25/504
Sec.
3]
Limits
6
179. Hm
(Vn
+
1
\f~n).
n
-+
<
-o/% i
/
180.
lim
When
seeking
the
limit
of
a ratio
of two
integral
polynomials
in
*
as
x
-+
oo,
it is
useful
first
to
divide
both
terms of
the
ratio
by
x
n
,
where
n
is
the
highest
decree
of
these
polynomials.
A
similar
procedure
is
also
possible
in
many
cases
for
fractions
contain-
ing
irrational
terms.
Example
1.
lim
J2^-3)(3t-f^)(4A'-6)
_
lim
.
*
=.
lim
J
=
1.
Example
2.
181.
lim
^rrr.
*86.
lim
^~~^=J.
r
-.
or
*
~'
1
*
+
V
X*
-\-
\
182.
lim
^^.
187.
lim
00
*
-1
__ .
1-
1/
jc
183.
lim
.,
.
J
.
188.
lim
3*
+
7
O
^2 Y L
^
184.
lim
4-
h
->*
10-j-
A:
3
8v
+5*
189.
lirn
185.
lim
-r-r~c
-
*
^
5
190.
lim
Vx
+
Vx
If
P(A-)
and
Q
(x)
are
integral
polynomials
and P
(u)
+
or
Q (a)
then
the
limit
of
the
rational
fraction
lim
is
obtained
directly.
But
if
P(a)
=
Q(a)=0,
then
it
is advisable
to
camel
the
binomial
*
a
P
(x)
out
of
the
fraction
Q
once
or several
times.
Example
3.
lim
/'T
4
^
lim
* )
f
xf
??
Hm
^^4.
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8/9/2019 Receuil Problems in Mathematical Analysis D
26/504
26
Introduction
to
Analysis
_
\Ch.
I
101.
lim
^{.
198.
Um
*
^
192.
lim
*-.
196.
lim
*
_.
|
*
^
*-+> fl
The
expressions
containing
irrational terms
are
in
many
cases
rational-
ized
by introducing
a
new
variable.
Example
4.
Find
lim
Solution.
Putting
+*
=
*/',
we
have
lim
E=1
Mm
^
= lim
2
3/
,
,,.
,'~
t/x
~
]
x
200.
lim
T
*~
.
199.
lim
-4^-.
201.
lim
X
-
1
*
l
Another
way
of
finding
the
limit of
an
irrational
expression
is
to
trans-
fer the
irrational
term from
the numerator
to
the
denominator,
or vice
versa,
from
the denominator
to
the
numerator.
Example
5.
lim
=
=
lim
_
^
x
-+a(
X
a)(Vx
+
V
a)
lim
*->
a
^
jc
-f
V
a
2\f~i
203.
lim
Q
-.
206.
lim
-=f.
-49
__
204.
li.n
j-^=
.
207.
lim
*-*
/
x 2
*-+<
205.
lim
^L
1
.
208.
lim
*-+'
/
* 1
^-*o
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8/9/2019 Receuil Problems in Mathematical Analysis D
27/504
Sec.
#]
Limits
27
209.
lim
_
K
210. lim
211.
Hm(]/xfa
|
Jf--fCO
The
formula
212.
lim
[/*(*
+
a)
xj.
X-
213 '
^i
214.
li.
-6*
4
6-*).
215.
llm
-i
X
-
X
r
frequently
used
when
solving
the
following
examples.
It is
taken
for
granted
that
lim
sin
*
=
sin
a
and
lim
cos
*=
cos
a.
Example
6.
216.
a)
lim;
217.
lira
lim
b)li.n^.
X
-> CO
,.
sill
3x
sin 5*
sin
2*
'
sin
JTX
218.
lim
X
-0
219.
lim
=
.
M
^
l
sin BJIJC
220.
lim
(
n
sin-).
n-*cc
\
n
I
221.
lim
222.
lim
223.
lim
224.
lim
225.
lim
226.
lim
crs^
227.
a)
lim
xsinl;
b)
lim
x
sin
.
X-*
00
*
228.
lim
(1
x)
tan
-~-
.
Jt-M
^
229.
lim
cot
2x
cot
f-^ x).
* -+0
\
*
/
230. lim
*-
Jt
231.
lim
ji
*
1-2
232.
lim
cosmx
-
cosn
\
*
*
233. lim
JC
-
tan
A:
sui
arc
sin ^
236.
lim
I
tan*
'
sin
six
'
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28
_
Introduction
to
Analysis
_
[Ch.
1
nx
m.
ta
.=T.
24
-
?.
n
* r
'
f
-*
1
I
p
Jt
When
taking
limits
of
the
form
lim
l(x)=
1
+a(x),
where
a
(x)
-*
as
x
-+
a
and,
lien^e,
1
Hm a
(x)
^
(x)
Hm
[(p
(x)
-
ij
ty
(x)
x
-
a
where
e
=
2.718 ...
is
Napier's
number.
Example
7.
Find
lim
Solution.
Here,
lim
(5111=2
and
lim
Jf-^O
\
X
hence,
lim
x-*o
Example
8.
Find
Solution.
We
have
lim
1
r-^(
2
J^
2
x
end
Hm
*
2
=
-
8/9/2019 Receuil Problems in Mathematical Analysis D
29/504
^Sec
3]
Limits
Therefore,
lim /
j
=0.
Example
9.
Find
lim
f
x
~~
\
t
Solution.
We
have
lim
^11=
lim
i
=
X-+
06
X
-4-
1
(- CO
.
,
1
+
T
Transforming,
as indicated
above,
we
have
In
this
case
it
is
easier
to
find the
limit
without
resorting
to
the
general
procedure:
Generally,
it
is useful
to
remember
that
lim
250.
li
244.
lim(*
2
*
M
].
251.
lirn(l
+
sinjc)
*.
X-K>\
X
3x4-2/
-.o
/^i
i
2
\*
a
J_
245>
Jill
(
2?+T
)
252**.
a)
lim
(cos
x)
*
;
/
1
\
X
~*
246.
Hmfl
-)
.
.
...
V
/
b)
H
247
Iim(l
f
I)*.
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30
_
Introduction to
Analysis
_
[Ch.
When
solving
the
problems
that
follow,
it
is
useful
to
know that
if the
limit
lim/(x)
exists
and
is
positive,
then
lim
[In
/(*)]
=
In
[Hm
f
(x)].
x-+a X-+Q
Example
tO. Prove
that
Solution.
We
have
lim
ln
X-*0
X
X-+Q
Formula
(*)
is
frequently
used
in
the
solution
of
problems.
253.
lim
[In
(2*+ )
*-
254.
li
.
-
X
255.
limfjlnl/J-i^).
260*.
llmn(^/a
\)
(a>0).
,_*
\
lX/
n
^
V)
pCLX ptX
256. lim
*[ln(jt+l)
Inx].
261.
lim-
-
.
*
0).
b)
lim
x
*
(see
Problems
103
and
104).
Find
the*
following
limits
that
occur
on
one
side:
264.
a)
lira
*_^
.
fa
Hm
i
b)Jirn*
p===.
* +
1+
'
T
265.
a/lLutanh*;
267
-
a
)
lim
*-*-*
*--
b)
limtanh*,
b)
Hm
*->+
*-*+
where
tanh^
=
^^~.
268.
a)
lim
266.
a)
lira
V
;
b)
|im
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Sec.
31
Limits
31
269.
a)
lim-^4i;
270.
a)
Hm-^-;
x
~
*
Construct
the
graphs
of
the
following
functions:
271**.
y
=
\im
(cos
2
*).
n->oo
272*.
y=lim
*
n
(x^O).
n-*c
*
i
x
273.
y
=
\im
J/V-t-a
2
.
a->o
274.
t/
=
li;n|
275.
t/
=
li
-*<
276.
Transform
the
following
mixed
periodic
fraction
into
a
common
fraction:
a
=
0.13555...
Regard
it
as
the
limit
of
the
corresponding
finite
fraction.
277.
What
will
happen
to the
roots
of
the
quadratic
equation
if
the
coefficient
a
approaches
zero
while
the coefficients b
and
c
are
constant,
and
fc^=0?
278. Find the
limit
of
the
interior
angle
of
a
regular
n-gon
as
n
>
oo.
279. Find
the
limit
of the
perimeters
of
regular
n-gons
inscribed
in
a circle
of
radius
R
and
circumscribed
about it
as n
-
oo.
20.
Find
the
limit
of
the
sum
of
the lengths
of
the
ordinates
of
the
curve
y
=
e~*cos
nx,
drawn
at
the
points
x
=
0,
1,
2,
...,
n,
as
n
*oo.
281.
Find the limit
of
the
sum
of
the
areas
of
the
squares
constructed
on
the
ordinates
of
the
curve
as
on
bases,
where
x=^l,
2,
3,
...,
n,
provided
that
n
*oo.
282. Find
the
limit
of
the
perimeter
of
a
broken
line
M^..
.M
n
inscribed
in
a
logarithmic
spiral
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Introduction
to
Analysis [Ch.
I
(as
n
oo),
if
the
vertices
of
this
broken
line
have,
respectively,
the
polar
angles
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Sec.
4]
Infinitely
Small
and
Large
Quantities
33
Sec. 4.
Infinitely
Small
and
Large
Quantities
1.
Infinitely
small
quantities
(infinitesimals).
If
lim
a
(x)
=
0,
x->a
i.e.,
if
|a(x)|tf.
then
the
function
f(x)
is
called
an
infinite
as
x
>a.
2-1900
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34
_
Introduction
to
Analysis
_
[Ch.
1
The
definition
of an
infinite
f
(x)
as
x
>
co
is
analogous.
As in
the
case
of
infinitesimals,
we
introduce
the
concept
of
infinites
of
different
orders.
288.
Prove
that
the function
is
an
infinitesimal
as
x
*oo.
For what
values of
x
is
the
ine-
quality
l/WI1.
For what
values
of
x is
the
ine-
quality
/(*) <
fulfilled
if
e
is
an
arbitrary
positive
number?
Calculate
numeri-
cally
for:
a)
e-0.1;
b)
e
=
0.01;
c)
e
=
0.001.
290.
Prove
that the
function
~
x 2
is
an
infinite for
x
*2.
In
what
neighbourhoods
of
|x
2|N
fulfilled
if
N
is
an
arbitrary
positive
number?
Find
5
if
a)
#=10;
b)
#=100;
J2^
c)
#=1000.
o
291.
Determine
the
order
of
smallness
of:
a)
the
surface
of
a
sphere,
b)
the
volume
of
a
sphere
if
the
radius
of the
sphere
r
is an infinitesimal
of
order
one.
What
will
the
orders
be
of the
radius
of
the
sphere
and
the volume of
the
sphere
with
respect
to
its
surface?
292. Let
the
central
angle
a
of
a
cir-
cular
sector
ABO
(Fig.
9)
with
radius
R
tend
to zero.
Determine
the
orders
of
the
infinitesimals
relative to the
infinitesimal
a:
a)
of
the
chord
AB\
b)
of
the
line
CD;
c)
of
the
area
of
A/4BD.
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Sec.
4]
Infinitely
Small
and
Large
Quantities
35
293.
For
x
*0
determine
the
orders of
smallness
relative
to
x
of
the functions:
*\
^*
d)
1
cos
*'*
*)
\
+x
e)
tan
A:
sin
A:.
b)
c)
$/*'-
294.
Prove that
the
length
of an
infinitesimal
arc
of
a circle
of constant radius
is
equivalent
to
the
length
of
its chord.
295.
Can
we
say
that
an
infinitesimally
small
segment
and
an
infinitesimally
small semicircle constructed
on
this
segment
as
a
diameter
are
equivalent?
Using
the
theorem
of
the
ratio
of
two
infinitesimals,
find
296.
lim
si
3
*'
s
5
*
.
298.
lim^
.
arc
sin
_^
=
299.
lim
297.
lim
,
f
1
-*
~
x
^o
ln(l--*)
300.
Prove that
when x *0 the
quantities
~
and
Y\
+xl
are
equivalent.
Using
this
result,
demonstrate that
when
\x\
is
small
we
have the
approximate
equality
VT+T1
+
.
(1)
Applying
formula
(1),
approximate
the
following:
a)
1/L06;
b)
1/0^7;
c)
/lO;
d)
/T20
and
compare
the
values obtained with tabular
data.
301.
Prove
that
when
x
we
have
the
following
approxi-
mate
equalities
accurate
to
terms
of
order
x
2
:
b)
c) (1
+x)
n
&\
+
nx
(n
is a
positive
integer);
d) log(l+x)
=
Afx,
where Af
=
log
e
=
0.43429...
Using
these
formulas,
approximate:
*>
02
;
2
>
0^7
;
3
>
I
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36
_
Introduction
to
Analysis
_
[Ch.
1
302.
Show
that
for
X+OQ
the
rational
integral
function
P
(x)
=
a.x
n
+
a,x
n
~
l
+
.
.
.
-f
a
n
(
is
an
infinitely
large
quantity equivalent
to
the
term
of
highest
degree
a
x
n
.
303.
Let
x*oo.
Taking
x to
bean
infinite
of
the
first
order,
determine
the order of
growth
of
the
functions:
a)
*>-
100*
-1,000;
c)
b)
7+2-
Sec.
5.
Continuity
of
Functions
1.
Definition
of
continuity.
A
function
/ (x)
is
continuous
when
x
=
(or
at
the
point
g ),
if:
1)
this
function
is
defined
at
the
point
g,
that
is,
there exists
a
number
/
(g);
2)
there
exists
a finite
limit
lim
f
(x);
3)
this lim-
x-4
it is
equal
to the value of
the
function
at the
point
g,
i.e.,
llmf
(*)
=
/().
(1)
*-*fc
Putting
where
Ag
^0,
condition
(1)
may
be
rewritten
as
lim
A/(g)
=
lim
l/(g+
Ag)-f
(g)]
=
0.
(2)
or
the
function
/ (x)
is continuous
at
the
point
g
if
(and
only
if)
at
this
point
to
an
infinitesimal
increment
in
the
argument
there
corresponds
an
infinitesi-
mal
increment in the
function.
If
a
function is
continuous
at
every point
of some
region
(interval, etc.),
then
it
is said
to be
continuous
in
this
region.
Example
1.
Prove
that
the
function
y
=
sin
x
fs
continuous for
every
value
of
the
argument
x.
Solution.
We
have
sin
Ay
=
sin
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Sec.
5]
Continuity
of
Functions
37
2.
Points of
discontinuity
of
a
function.
We
say
that
a
function
/(x)has
a
discontinuity
'at
x=*
(or
at
the
point
X
Q
)
within
the
domain
of
definition
of
the function or on
the
boundary
of this
domain if
there
is
a
break
in
the
continuity
of
the
function
at
this
point.
Example
2.
The
function
f(x)=
(Fig.
10
a)
is
discontinuous
when
x=l.
This
function
is
not
defined at the
point
x
1,
and
no
matter
1
2
1-2
how
we
choose
the
number
/(I),
the
redefined
function
/
(x)
will
not
be
con-
tinuous
for
*=1.
If
the
function
f
(x)
has
finite
limits:
Hm
/(*)
=
f(*
-0)
and
Urn
/(*)
=
/(
and
not all
three
numbers
f(x
),
/(*
)
f
(x
+
Q)
are
equal,
the
nx
Q
is
called
a
discontinuity
of
the
first
kind.
In
particular,
if
then
*
is called
a
removable
discontinuity.
For
continuity
of
a function
f(x)
at
a
point
JC
Q
,
it is
necessary
and suf-
ficient
that
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38
In
t
reduction
to
Analysis
[Ch.
I
Example
3.
The
function
/(jc)=j-y
has
a
discontinuity
of the
first
kind
at
*
=
0.
Indeed,
here,
/ (
+
0)=
lim
5 L
==+
i
and
/(_0)=
lim
jc-*-o
x
Example
4.
The
iunction
y
=
E(x),
where
E(x)
denotes
the
integral
part
of
the
number
x
[i.e.,
E
(x)
is an
integer
that
satisfies
the
equality
x
=
E(x)
+
q.
where
0
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Sec.
5]
_
Continuity
of
Functions
_
39
305. Prove
that
the
rational
integral
tunction
is
continuous
for
any
value
of
x.
306.
Prove
that
the
rational
fractional
function
is
continuous
for all
values
of
x
except
those
that
make
the
de-
nominator
zero.
307*.
Prove
that
the
function
y
=
Yx
is
continuous
for
x&zQ.
308.
Prove
that
if
the
function
f (x)
is continuous
and non-
negative
in
the
interval
(a,
6),
then
the
function
is
likewise
continuous in
this
interval.
309*.
Prove
that
the
function
y
cos
x
is
continuous
for
any
x.
310.
For
what
values
of
x
are
the
functions
a)
tan*
and
b)
cotjc
continuous?
311*.
Show
that
the
function
#
=
|#|
is
continuous.
Plot
the
graph
of
this
function.
312.
Prove
that the absolute
value
of a
continuous
function
is
a continuous function.
313.
A
function
is
defined
by
the
formulas
How
should
one
choose
the value
of the
function
A=f(2)
so
that
the
thus
redefined
function
f(x)
is
continuous
for
#
=
2?
Plot the
graph
of
the function
y
=
f(x).
314.
The
right
side
of the
equation
f(x)
=
lx
sin
is
meaningless
for x
=
0.
How
should
one
choose the value
/(O)
so
that
f(x)
is
continuous for
jc
=
0?
315. The
function
f(*)
=
arctan--^
is
meaningless
for
x=--2. Is
it
possible
to
define
the
value
of
/(2)
in
such a
way
that
the redefined
function
should be continuous
for jc
=
2?
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40 Introduction
to
Analysis
[Ch.
1
316.
The function
f(x)
is
not defined
for
x
=
0.
Define
/(O)
so
that
fix)
is
continuous
for
x
=
0,
if:
/]
I
y\__1
a)
f(x)
=
l
^
y
;
-
(n
is
a
positive
integer);
b)
/(*)
=
c)
/(*)
=
d)
/(
x
1
cos*
.
A*
ln(\+x)
111(1
f)
/(*)
=
* cot*.
Investigate
the
following
functions
for
continuity:
317.
y
=
-.
324.
3.
331.
Prove that
the Dirichlet
function
%(x)
t
which
is
zero
for
irrational
x
and
unity
for
rational
x,
is
discontinuous
for
every
value
of x.
Investigate
the
following
functions for
continuity
and
construct
their
graphs:
332.
y
=
\in\
333.
y
=
lim
(x
arc
tan
nx).
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Sec.
5]
Continuity
of
Functions
334.
a)
y
=
sgnx, b)
y
=
x
sgnx,
c)
i/
=
sgn(sinjt),
where
the
function
sgn
x
is
defined
by
the
formulas:
I
+
1,
if
*>0,
sgn
x
=
{
0,
if
x=
0,
[
-1,
if
*
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Chapter
II
DIFFERENTIATION
OF
FUNCTIONS
Sec.
1.
Calculating Derivatives
Directly
1. Increment of
the
argument
and
increment
of the function.
If
x
and
x
l
are values
of
the
argument
x,
and
y
=
f(x)
and
t/
1
=
/(jc
1
)
are
corresponding
values
of
the
function
y
=
f(x),
then
^x~x
l
x
is
called the
increment
of
the
argument
x
in
the
interval
(x,
xj,
and
A0=0i
y
or
/
=
f
(x,)
-f
(x)
=
f
(x
+
A*)
-
r
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Sec.
1]
Calculating
Derivatives
Directly
43
calculate
Ax
and
A#,
corresponding
to
a
change
in
the
argument:
a)
fromx=l
to
x=l.l;
b)
from
x=3
to
x
=
2.
Solution.
We
have
a)
Ax=l.
1
1=0.1,
Ai/
=
(l.l
2
5-1.1
+
6)
(I
2
5-
1+6)
=
0.29;
b)
Ax
=
2
3
=
1,
At/
=
(2*
5-2-1-6)
(3*
5-3
-f-
6)--=0.
Example
2.
In
the
case
of
the
hyperbola
y
=
,
find
the
slope
of
the
secant
passing
through
the
points
M
(
3,
--
)
and
N
{
10,
-r^ )
.
V
'
1 1
J
7
Solution.
Here,
Ax=10
3
=
7
and
Ay
=
^
4=
5*-
Hence,
1U
o
5U
,
AJ/
1
Ax~~
30'
2.
The
derivative.
The
derivative
y'=j-
of
a
function
y-=f(x)
with
re-
spect
to
the
argument
x is
the
limit
of
the
ratio
-r^
when
Ax
approaches
zero;
that
is.
y>=
lim
>.
AJC
->
o
A*
The
magnitude
of
the
derivative
yields
the
slope
of
the
tangent
MT to
the
graph
of the
function
y
=
f(x)
at
the
point
x
(Fig.
11):
y'
tan
q>.
Finding
the
derivative
/'
is
usually
called
differentiation
of
the
function.
The
derivative
y'=f'
(x)
is
the rate
of
change
of
the
function
at
the
point
x.
Example
3.
Find
the derivative of the function
y
=
x*.
Solution.
From
formula
(1)
we
have
Ay
=
(*+ A*)*
x
i
2*Ax+
(Ax)
1
and
Hence,
5*.
One-sided
derivatives.
The
expressions
/'_(*)=
lim
f
(*+**)-/(*)
AJ:-*--O
Ax
and
/(x)=
lim
'=
lim
L^
lim
Ax
AJC->O
Ax
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44
'
Differentiation
of
Functions
[Ch.
2
are
called,
respectively,
the
left-hand
or
right-hand
derivative
of
the
function
f(x)
at the
point
x.
For
/'
(x)
to
exist,
it
is
necessary
and
sufficient
that
/'.(*)
=
/+(*).
Example
4
Find
/'_ (0)
and
/'
+
(0)
of
the
function
Solution.
By
the
definition
we
have
/'_ (0)
=
lim
L
f^
(0)
=
lim
A*--t-o
Ax
4.
Infinite
derivative.
If
at
some
point
we
have
Uoo,
im
/(*+**)-/(*)_.
then
we
say
that the
continuous
function
/
(x)
has
an
infinite
derivative
at
x.
In
this
case,
the
tangent
to
the
graph
of the
function
y
=
f(x)
is
perpendicu-
lar
to
the
x-axis.
Example
5.
Find
/' (0)
of
the
function
V=V*
Solution. We
have
/'0)=llm
*~
=
]im
-=-=
-
8/9/2019 Receuil Problems in Mathematical Analysis D
45/504
Sec.
1]
Calculating
Derivatives
Directly
45
345. Find
Ay
and
-
which
correspond
to
a
change
in
argu-
ment fromx
to
x-(-
Ax
for the
functions:
a)
y-ax
+
6;
d)
y
=
/x;
b)
y-x'; e)
y
=
2*\
346.
Find
the
slope
of the
secant
to the
parabola
y
==
~x
x
t
if
the
abscissas
of
the
points
of
intersection
are
equal:
a)
x,-l,
x
a
-2;
c)
x^l,'
x
2
2
=l+fc.
To
what limit
does
the
slope
of the
secant tend
in
the
latter
case
if
/i->0?
347.
What
is
the
mean
rate of
change
of the
function
y
=
x*
in
the interval
l^x^4?
348.
The
law
of motion
of
a
point
is
s
=
2/
2
+
3/
+ 5,
where
the
distance
s
is
given
in
centimetres
and
the
time
t
is
in
seconds.
What
is
the
average velocity
of
the
point
over
the
interval
of
time from t~\
to
^
=
5?
349.
Find
the mean
rise of the
curve
y
=
2*
in the
interval
350.
Find
the
mean rise
of
the
curve
j/
=
/(x)
in the
interval
[x,
x+Ax].
351.
What is
to
be
understood
by
the
rise of
the
curve
y
=
f(x)
at
a
given
point
x?
352.
Define:
a)
the
mean
rate
of
rotation;
b)
the
instantaneous
rate
of
rotation.
353.
A
hot
body
placed
in
a
medium of
lower
temperature
cools off.
What
is to
be
understood
by:
a)
the
mean rate
of
cooling;
b)
the
rate of
cooling
at
a
given
instant?
354.
What
is
to
be understood
by
the
rate
of reaction
of
a
sub-
stance
in
a
chemical
reaction?
355.
Let
M
=
/(X)
be
the mass
of
a non-
homogeneous
rod
over
the interval
[0,
x].
What
is to
be
understood
by:
a)
the
mean
linear
density
of
the rod on
the interval
[x,
x+Ax];
b)
the
linear
density
of
the rod
at a
point
x?
356.
Find
the
ratio
of
the
function
*/
=
at
the
point
x
=
2,
if:
a)
Ax-1;
b)
Ax
=
0.1;
c)
Ax -0.01.
What
is the deriv-
ative
y'
when x^2?
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8/9/2019 Receuil Problems in Mathematical Analysis D
46/504
46
_
Differentiation
of
Functions
_
[C/t.
2
357**.
Find
the
derivative
of
the
function
y
=
ianx.
358.
Find
{/'=
lirn
^
of
the
functions:
a)
t/
=
x
f
;
c) y
=
359.
Calculate
f'(8),
if
360. Find
/'(0),
/'(I), /'(2),
if
/(*)
=
*(*-
1)
1
(x-2)V
361.
At what
points
does
the
derivative of
the
function
/(#)
=
#*
coincide
numerically
with
the
value
of
the
function
itself,
that
is,
/(*)
=
/'(*)?
362.
The
law
of
motion
of
a
point
is
s
=
5/*,
where
the
dis-
tance
s
is
in
metres
and
the
time t
is
in
seconds. Find
the
speed
at
*
=
3.
363. Find
the
slope
of the
tangent
to the
curve
y
=
Q.lx*
drawn
at
a
point
with
abscissa
x
=
2.
364. Find
the
slope
of
the
tangent
to
the
curve
y=sinjt
at
the
point
(ji,
0).
365.
Find
the
value of the
derivative
of the
function
f
(*)
=
-i
i
\
/
x
at
the
point
x
=
X
Q
(x
+
0).
366*.
What
are
the
slopes
of
the
tangents
to
the
curves
y
=
~
and
y
=
x*
at
the
point
of their
intersection?
Find
the
angle
be-
tween
these
tangents.
367**. Show
that
the
following
functions
do
not
have
finite
derivatives
at
the
indicated
points:
a)
y=^?
_
at
x
b)
y
=l/xl
at
x
c)
y
=
|cosx|
at
*
=
jt,
fc
=
0,
1,
2,
Sec.
2.
Tabular
Differentiation
1.
Basic rules for
finding
a
derivative. If
c
is
a
constant
and
w
=
o>(jc)
v
ty(x)
are
functions
that
have
derivatives,
then
'
1) (c)'
=
0;
5)
2)
(*)'=,;
6)
3) ( )'-'
t;';
7)-
=
=
(v *
0).
4)
(cu)'=cu
r
;
-
8/9/2019 Receuil Problems in Mathematical Analysis D
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-
8/9/2019 Receuil Problems in Mathematical Analysis D
48/504
48
_
Differentiation
of
Functions
_
[Ch.
2
Example
1.
Find
the
derivative of the
function
Solution.
Putting
#
=
a
5
,
where
w
=
(*
2
2jc
+
3), by
formula
(1)
we
will
have
y'
=
(u*)'
u
(*
2
-2x
+
3);
=
5u
4
(2x-2)
=
10
(x-1)
(jt
2
-
Example
2.
Find
the
derivative
of
the
function
y
=
sin
8
4*.
Solution.
Putting
(/
=
*;
u
=
sinu;
u
=
4jc,
we
find
>-4
=
12sin
2
4xcos4jt.
Find the derivatives
of
the
following
functions
(the
rule
for
differentiating
a
composite
function is not
used
in
problems
368-408).
A.
Algebraic
Functions
s
5
QO
,.
V
5 A
V
8
|
O
v
O
Q7
11
Q v
3
O
v
2
I v~'
ouo.
y
A
~~
TEA
~]~
AA
*j.
Of
i/.
t^
JA
~~~
^A.
~f~
A
.
Q5Q
^
v I v^
A V
4
Q7IS4T f
v*
*
/ +f*
oOy.
i/
==:
' :
;r-Jt-pJT
U.OA
.
o/O .
y
=
X
y
X .
370.
y--=ax*
-f-
&A:
+
C.
372
(,-a(-H-W .
379.
il-
373.
if
.
380.
=
.
V^a
2
+
6
2
2jc
1
JC
374.
y==+
nn
2.
381.
=
i
B.
Inverse
Circular
and
Trigonometric
Functions
382.
(/
=
5
sin
^
+
3
cos
x.
386.
y=arctan^-h
arc
cot x.
383.
t/
=
tanx
cotx.
387.
f/
=
388
-
-
S85.j/-2/sin(-(''-2)cos(.
389.
_('+ ')'
' '-'.
^
-
8/9/2019 Receuil Problems in Mathematical Analysis D
49/504
Sec.
2]
_
Tabular
Differentiation
_
49
C.
Exponential
and
Logarithmic
Functions
390.
y^K*-e*.
396.
y
=
e*arc
sin
x.
391.
y
=
(x-l)e*.
397.
y^
~.
392.
r/
=
5-
398
-
y
=
*
.
\j
393.
(/==
J.
399.
*/
=
7
394.
/
(x)
=
e*
cos
jc.
400.
y
=
\nx\ogx
In a
log
a
jc.
395.
#=:(A;
2
2
D.
Hyperbolic
and
Inverse
Hyperbolic
Functions
401.
t/
=
Jtsinhjt.
405.
(/
=
arctanx
arctanh
402.
y=-V-
.
406.
t/
=
arc
cosh
x
J
403. //
=
tanhA:
*.
407.
(/
=
-
404
t/
=
^iiL
408.
//
=
-
^
Inx
^
i
x-
E.
Composite
Functions
In
problems
409
to
466,
use
the
rule
for
differentiating
a
composite
func-
tion
with one
intermediate
argument.
Find the
derivatives
of
the
following
functions:
40Q** a H -i_^v ^r
2
\*
v
u
.
y
\
i
i~
\jAt
~~~-
*jAi
j
Solution.
Denote
1
+
3jt 5jc*
=
w;
then
t/
=
w
j
f/
~56(2*
1)'
24(2^1)'
40(2x
I)''
414.
t/=J/T^J? .
415.
y=^/
416.
w=(a''.
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8/9/2019 Receuil Problems in Mathematical Analysis D
50/504
50
Differentiation
of
Functions
_
[Ch.
2
417.
t/
=
(3
2
sin*)
5
.
Solution.
y'
=
5
(32
sin
Jt)
4
-(3
2
sin
x)'
=
5
(3
2
sin
x)* (
2 cos
x)
=
-
10
cos
x
(3
2 sin
x)
4
.
418.
j/=tanjc
-
t
419.
r/=J/coU
/coU.
423.
j/
=
0-^-3
--
y
3
cos
3
*
CGSJC
420.
y
=
2x
+
5
cos'
*.
424.
y
=
|/
3sin*--2cos*
421*. x
=
cosec
2
^+sec
f
/.
425.
y=
422.
f(x)
=
6(1
_
3cosx)
f
426.
{/=
1/1
+
arc
sin x.
427.
y
=
J/arc
tan *
(arc
sin
x)
9
.
428 -
y
429.
t/
430.
y=/2e
x
431.
y
=
sin
3*
+
cos
-|-
+
tart
Solution
,
f^
=
cos
3^3*)'
-sin
4
f
4V
+
,
]
/-
5
\
5
/
cos
2
Y
x
top related